Abstract
In this paper, the concept of discontinuity geometry of rigid impact oscillators is extended to soft impact oscillators and the mechanisms of grazing bifurcations of an impact oscillator with one-sided elastic constraint are studied by the geometric and dynamical systems methods. The existence conditions of periodic solutions are given by the discontinuity geometry objects and then are used to derive the discontinuity curves. Several bifurcation scenarios near grazing bifurcation are studied to examine the evolution of the system dynamics from a geometrical point of view. The geometrical conditions are obtained for the existence of periodic orbits with one impact, grazing and saddle-node bifurcations. Some geometrical insight is gained into a question whether there is a discontinuous jump or a continuous transition from a non-impacting period-1 to an impacting period-1 attractor at a grazing bifurcation.
| Original language | English |
|---|---|
| Pages (from-to) | 662-678 |
| Number of pages | 17 |
| Journal | IMA Journal of Applied Mathematics |
| Volume | 81 |
| Issue number | 4 |
| Early online date | 8 Mar 2016 |
| DOIs | |
| Publication status | Published - 2016 |
Bibliographical note
FundingThis work is partially supported by the National Natural Science Foundation of China (Grant Nos 11402224, 11202180, 61273106, 11171290), the Qin Lan Project of the Jiangsu Higher Education Institutions of China, and the Jiangsu Overseas Research and Training Program for University Prominent Young and Middle-aged Teachers and Presidents.
Acknowledgements
H.J. acknowledges the hospitality of the University of Aberdeen. We also thank Prof. Yoshisuke Ueda and Dr Yang Liu for helpful comments.
Keywords
- Discontinuity geometry
- Grazing induced bifurcations
- Impact oscillators
- Non-smooth systems